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Diophantine representation of the decimal expansions of \(e\) and \(\pi \). (English) Zbl 0984.11007
Summary: Let \(\alpha \in \{\operatorname {e},\pi \}\), \(\alpha =[\alpha]+\sum _{\kappa =1}^\infty \alpha _\kappa(\beta)\cdot \beta ^{-\kappa}\) (where \(\beta \in \mathbb N\setminus \{1\}\) and \(\alpha _\kappa(\beta)\in \{0,1,\dots ,\beta -1\}\)) and \(\zeta \in \{0,1,\dots ,\beta -1\}\). We describe short Diophantine representations for the predicate \(\alpha _\kappa(\beta)=\zeta \). The proofs use methods that were developed for the solution of Hilbert’s Tenth Problem.
11A63 Radix representation; digital problems
11U05 Decidability (number-theoretic aspects)
Full Text: EuDML
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